Multiple colors on a single graph edge?

trg = TransitiveReductionGraph[RelationGraph[UnsameQ@## && SubsetQ@##&, Subsets[Range@3]], VertexLabels -> "Name", EdgeShapeFunction->"Line"] 

SetProperty[trg, {EdgeStyle -> {DirectedEdge[{1, 2} | {1}, _] :> Directive[Thick, Red], DirectedEdge[{2, 3} | {3}, _] :> Directive[Thick, Blue]}, EdgeShapeFunction -> {DirectedEdge[{2}, _] :> ({Red, Thick, Translate[GraphElementData["Line"][##], {-.015, 0}], Blue, Translate[GraphElementData["Line"][##], {.015, 0}]} &)}}] 

Alternatively, you can have two curved lines for the edge {2} -> {}:

SetProperty[trg, {EdgeStyle -> {DirectedEdge[{1, 2} | {1}, _] :> Directive[Thick, Red], DirectedEdge[{2, 3} | {3}, _] :> Directive[Thick, Blue]}, EdgeShapeFunction -> {DirectedEdge[{2}, _] :> ({Red, Thick, GraphElementData[{"CurvedArc", "Curvature"-> -1/4}][##], Blue, GraphElementData[{"CurvedArc", "Curvature"-> 1/4}][##]} &)}}] 

I have a method to achieve what you want but the procedure is very dirty:

We will make two graphs and then Overlay them.

The first one is as shown below:-

 << Combinatorica`; g = MakeGraph[ Subsets[Range[3]], ((Intersection[#2, #1] == #1) && (#1 != #2)) &, VertexLabel -> True]; hdg = HasseDiagram[g]; sgX = ShowGraph[ Highlight[ hdg, {{{3, 1}}, {{1, 2}, {2, 5}, {5, 3}, {3, 1}}, {{1, 4}, {4, 7}, {7, 3}, {3, 1}}}, HighlightedEdgeColors -> {Blue, Red, Blue}, HighlightedEdgeStyle -> Thickness[0.008]], ImageSize -> 300, VertexStyle -> PointSize[0.025], EdgeStyle -> Thin, VertexNumber -> True, VertexNumberColor -> Gray] 

The second graph needs to have only one red line so that we can overlay the first on top of the second. To make all other edge invisible I'm coloring them white. Rest of the elements (vertex,label) are also suppressed.

 g1 = MakeGraph[ Subsets[Range[3]], ((Intersection[#2, #1] == #1) && (#1 != #2)) &, VertexLabel -> False] hdg1 = HasseDiagram[g1]; sgX1 = ShowGraph[ Highlight[ hdg1, {{{3, 1}}, {{6, 4}, {4, 7}, {7, 8}, {8, 6}}, {{2, 5}, {5, 8}, {8, 6}, {6, 2}}, {{1, 4}, {4, 7}, {7, 3}, {3, 1}}, {{1, 2}, {2, 5}, {5, 3}}}, HighlightedEdgeColors -> {Red, White, White, White, White}, HighlightedEdgeStyle -> Thickness[0.008]], ImageSize -> 300, VertexStyle -> PointSize[0.005], EdgeStyle -> Thin, VertexNumber -> False] 

Now we are ready to overlay them. Before that we will need to set some parameters as shown below.

{w, h} = {350, 400}; pt = Scaled[{0.5, 0.5}]; {{l, r}, {b, t}} = {{80, 40}, {60, 10}}; 

Now we Overlay the first image on top of second and then you can move the first image freely to achieve the final position.

 res = Overlay[{sgX1, Graphics[{}, AspectRatio -> (h + b + t)/(w + l + r), ImageSize -> {w + l + r, h + b + t}, ImagePadding -> 10, Epilog -> {Dynamic[Locator[Dynamic[pt], sgX]]}]}, All, 2] 

Once you have placed them properly then you can freeze the image as follows:

 p2 = res /. Locator[x_, y_] :> Inset[y, x] /. Dynamic :> Identity 

You may need to play around a little bit during the overlay process to get the two lines (red and blue) as close to each other as possible. Here is the final result

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Note: In your code the command HighlightedEdgeStyle should be inside Highlight